Swiatek, who counts the 2022 US Open among her five Grand Slam titles, converted three of six break-point chances but also racked up 41 unforced errors en route to securing the victory in one hour and 52 minutes at Arthur Ashe Stadium.
Rakhimova, who was added to the main draw as a lucky loser after another player pulled out with injury, nearly pushed the first-time meeting to a decider but squandered a trio of set points in the second-set tiebreak.
Swiatek built a 4-0 double break lead and looked poised to run away with an easy win but Rakhimova consolidated a break to get within a game before Swiatek went on to seal the first set with her third love hold of the match.
The Pole consolidated an early break with another love hold for a 3-1 lead in the second set but suddenly started showing signs of frustration as her shots were off target and Rakhimova refused to back down from the challenge.
Swiatek had a chance to serve out the match but Rakhimova broke to get to 5-5 as the second set went to a tiebreak where the Russian jumped ahead 6-3 before a slew of untimely mishits gifted her opponent the match.
"At the beginning (I felt) good but then I got a little bit tight and my opponent used it so I was trying to get back to my game," said Swiatek.
"I've just been trying to adjust to the courts, haven't, you know practised a lot so this time I just wanted to feel how it is on Arthur Ashe and I'm pretty sure day by day I am going to get more and more rhythm."
The reigning French Open champion will next face Japan's Ena Shibahara.
